To make the caramelized onions:

Place a large skillet over low heat and when it is hot, add the oil. Add the remaining ingredients and cook, stirring only occasionally as necessary, until the onions are deeply browned and slightly gooey, about 40 minutes.

If the onions dry out, add water, 1 tablespoon at a time. Use immediately or cover and refrigerate for up to 2 days. (Yield: ¾ to 1 cup)

To make the burgers:

Place the beef on a work surface and divide into 4 balls of equal size.

Divide the blue cheese into 4 portions of equal size.

Using your thumb or the back of a tablespoon, make a shallow hole in each portion of beef and fill it with the blue cheese. Shape the burger and carefully seal the opening.

Form each into a patty about ¾ to 1 inch thick, tossing it back and forth between your hands. To ensure more even cooking, make a ½-inch indentation with your thumb in the center of the burger. Handle the patties as little as possible; do not work more than necessary.

Prepare a grill to medium-high. When the coals are glowing red, after 15 to 20 minutes, cover with the grate. After 5 minutes, use a wire brush to thoroughly clean the grate. Brush the grate with oil. When the coals are covered with pale gray ash and you can leave your hand 5 inches above the fire for 2 to 3 seconds, the coals are ready.

Sprinkle both sides of the burgers with the salt and pepper. Place the burgers on the grate and grill until well seared on both sides, about 5 minutes for medium rare or 6 minutes for medium.

Top with a few crumbles of cheese and let melt, if desired. Transfer to buns (or toast as shown) or a serving platter and serve immediately.

Alternatively, place a cast iron skillet over high heat and when it is hot but not smoking add the burgers to the dry pan. Cook until well seared on both sides, 8 to 10 minutes for medium rare or 10 to 12 minutes for medium.